Question

Remainder when x^3+3x^2+3x+1 is divided by 2x+1 is

Answer 1

Step-by-step explanation:

Given:-

x^3+3x^2+3x+1

To find:-

Find the remainder when x^3+3x^2+3x+1 is divided by 2x+1 ?

Solution:-

Given cubic polynomial P(x) = x^3+3x^2+3x+1

Given divisor = 2x+1

We know that

Remainder Theorem:-

Let P(x) be a polynomial of the degree greater than or equal to 1 and (x-a) is another linear polynomial, if P(x) is divided by (x-a) then the remainder is P(a).

If x^3+3x^2+3x+1 is divided by 2x+1 then the remainder is P(-1/2) .

(Since , 2x+1=0=>x = -1/2)

The remainder = P(-1/2)

=> (-1/2)^3 +3(-1/2)^2 +3(-1/2) + 1

=> (-1/8) +3(1/4) +(-3/2) + 1

=> (-1/8)+(3/4)-(3/2)+1

LCM of 8 ,4 and 2 = 8

=> [(-1×1)+(3×2)-(3×4)+(1×8)]/8

=> (-1+6-12+8)/8

=> (14-13)/8

=> 1/8

Therefore, P(-1/2)=1/8

Answer:-

The required remainder for the given problem is 1/8

Used formula:-

Remainder Theorem:-

Let P(x) be a polynomial of the degree greater than or equal to 1 and (x-a) is another linear polynomial, if P(x) is divided by (x-a) then the remainder is P(a).